7-6 Similarity Transformations

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Presentation transcript:

7-6 Similarity Transformations p. 511 You identified congruence transformations. Identify similarity transformations. Verify similarity after a similarity transformation.

Definitions Transformation – an operation that maps an original figure (preimage) onto a new figure (image). Dilation – a transformation that enlarges or reduces the original figure proportionally. Center of dilation – a fixed point used in a dilation Scale factor of a dilation – the extent of the dilation. Scale factor – the ratio of a length on the image to a corresponding length on the preimage. The letter “k” usually represents the scale factor of a dilation. The value of “k” determines whether the dilation is an enlargement or a reduction. A dilation is a type of similarity transformation since it produces a similar figure.

p. 511

Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation. B is smaller than A, so the dilation is a reduction. The distance between the vertices at (2, 2) and (2, –2) for A is 4 and from the vertices at (1, 1) and (1, –1) for B is 2. Answer: So, the scale factor is or . _ 1 2 4

Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation. B is larger than A, so the dilation is an enlargement. The distance between the vertices at (3, 3) and (–3, 3) for A is 6 and from the vertices at (1, 1) and (–1, 1) for B is 2. Answer: So, the scale factor is or 3. _ 6 2

Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation. A. reduction; B. reduction; C. enlargement; 2 D. enlargement; 3 __ 1 3 2

B. Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation. A. reduction; B. reduction; C. enlargement; D. enlargement; 2 _ 1 3 2

PHOTOCOPYING A photocopy of a receipt is 1 PHOTOCOPYING A photocopy of a receipt is 1.5 inches wide and 4 inches long. By what percent should the receipt be enlarged so that its image is 2 times the original? What will be the dimensions of the enlarged image? To enlarge the receipt 2 times the original, use a scale factor of 2. Written as a percent, the scale factor is (2 ● 100%) or 200%. Now, find the dimensions of the enlarged receipt. width: 1.5 in. ● 200% = 3 in. length: 4 in. ● 200% = 8 in. Answer: The enlarged receipt will be 3 inches by 8 inches.

PHOTOGRAPHS Mariano wants to enlarge a picture he took that is 4 inches by 7.5 inches. He wants it to fit perfectly into a frame that is 400% of the original size. What will be the dimensions of the enlarged photo? A. 15 inches by 25 inches B. 8 inches by 15 inches C. 12 inches by 22.5 inches D. 16 inches by 30 inches

Verifying Similarity after a Dilation If you want to verify that a dilation produces a similar figure, you can compare corresponding angles and sides. For triangles, use SAS Similarity

A. Graph the original figure and its dilated image A. Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation. original: M(–6, –3), N(6, –3), O(–6, 6) image: D(–2, –1), F(2, –1), G(–2, 2) Graph each figure. Since M and D are both right angles, M  D. Show that the lengths of the sides that include M and D are proportional. Use the coordinate grid to find the lengths of the vertical segments MO and DG and the horizontal segments MN and DF. Answer: Since the lengths of the sides that include M and D are proportional, ΔMNO ~ ΔDFG by SAS Similarity.

B. Graph the original figure and its dilated image B. Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation. original: G(2, 1), H(4, 1), I(2, 0), J(4, 0) image: Q(4, 2), R(8, 2), S(4, 0), T(8, 0) Since the figures are rectangles, their corresponding angles are congruent. Find and compare the ratios of corresponding sides. Answer:

A. Graph the original figure and its dilated image A. Graph the original figure and its dilated image. Then determine the scale factor of the dilation. original: B(–7, –2), A(5, –2), D(–7, 7) image: J(–3, 0), K(1, 0), L(–3, 3) A. B. C. D. __ 1 2 3 4

B. Graph the original figure and its dilated image B. Graph the original figure and its dilated image. Then determine the scale factor of the dilation. original: A(4, 3), B(6, 3), C(4, 2), D(6, 2) image: E(6, 4), F(10, 4), G(6, 2), H(10, 2) A. 2 B. C. 3 D. 4 __ 1 3

7-6 Assignment Page 514, 6-13, 15